Palm Springs International Airport (PSP), Ontario International Airport (ONT), Los Angeles International Airport (LAX), San Diego International Airport (SAN)
Let's just be frank: there is no great airport option for Coachella. The Palm Springs airport is by far the most convenient, but ungodly expensive during Coachella. It's a really small airport with very limited flights. If you book early you can snag a good deal to Ontario airport (especially on Southwest), which isn't too far. More than likely you will find the best price to either Los Angeles or San Diego. Insider Tip: I always fly into San Diego. Everyone thinks Los Angeles is so much closer, but they are near equidistant, and I find flying into San Diego much more pleasant, and often cheaper. At least check both options to see what has the better deal for you. With either option you are looking at about a 2.5 hour drive from the airport to wherever you are staying for Coachella. From either location it is an easy drive and highway the entire way, but it is quite a hike. Note that Coachella does offer some shuttle options from LAX you can look into if you are against renting a car.
Essentially it will be the cost of renting a car, which will be roughly $40-$50 date. If you skip the car and use the Coachella shuttles from LAX they begin at $70 to the Coachella campgrounds. Just notice these have pretty strict hours so you need to plan around their schedule.
Generally the cheapest time to book flights for Coachella will be about six weeks before Coachella. This means you should be focused on booking your flight in March 2, 2018. This may seem late to you planners out there, but statistically it's the cheapest time to book flights. That said, if this is too late for you book it earlier. One final note is that plane tickets for Coachella tend to skyrocket about two weeks out, so certainly don't wait that long to book a Coachella flight.